Proposition 8

— Suppose that E is Hausdorff; let $\mu$ be a measure on X. The mapping $f \mapsto \int f d\mu$ of $\mathscr{K}(X;E)$ into $\widehat{E}$ (No. 3, Cor. 1 of Prop. 7) is the unique continuous linear mapping $\Phi$ such that $\Phi(g \cdot a) = \mu(g) \cdot a$ for every vector $a \in E$ and every function $g \in \mathscr{K}(X;\mathbf{C})$.

To prove the continuity of the mapping $f \mapsto \int f d\mu$, it suffices to show that its restriction to $\mathscr{K}(X,K;E)$ is continuous for every compact subset K of X (TVS, II, §4, No. 4, Prop. 5). We note that if the topology of E is defined by a family of semi-norms $(q_\alpha)$, that of $\mathscr{K}(X,K;E)$ is defined by the family of semi-norms

Now, let $h$ be a continuous mapping of X into $[0,1]$, with compact support and such that $h(x) = 1$ on K; by No. 2, Prop. 6 we have, for every function $f \in \mathscr{K}(X,K;E)$,

(the $q_\alpha$ being extended by continuity to $\widehat{E}$), which proves the continuity of $f \mapsto \int f d\mu$. On the other hand, with the notations of the statement,

by virtue of No. 1, Example 1 and Prop. 2 of No. 2 applied to the canonical injection $\mathbf{C} \cdot a \to E$. Moreover, the subspace of $\mathscr{K}(X;E)$ formed by the linear combinations $\sum_i g_i \cdot a_i$, where $a_i \in E$ and $g_i \in \mathscr{K}(X;\mathbf{C})$, is dense in $\mathscr{K}(X;E)$ (§1, No. 2, Prop. 5), which completes the proof.